We introduce the normal and anomalous Green's functions in the Matsubara representation:[1]}
\(&& G_{nn^{\prime },ss^{\prime }}( {k} ,\tau )=-\langle T_\tau c_{ {k} ,n,s}(\tau )c^\dagger _{ {k} ,n^{\prime },s^{\prime }}(0)\rangle , \nonumber \\&& F_{nn^{\prime },ss^{\prime }}( {k} ,\tau )=\langle T_\tau c_{ {k} ,n,s}(\tau )\tilde{c}_{ {k} ,n^{\prime },s^{\prime }}(0)\rangle , \\&& \bar{F}_{nn^{\prime },ss^{\prime }}( {k} ,\tau )=\langle T_\tau \tilde{c}^\dagger _{ {k} ,n,s}(\tau )c^\dagger _{ {k} ,n^{\prime },s^{\prime }}(0)\rangle . \nonumber \) 
